Video: The Components of a Vector on a Grid

The components of the vector 𝑢 are 〈−2, 1〉 as the terminal point of the vector is −2 units right (2 units left) of the initial point and 1 unit up from the initial point. What are the components of the vector 𝑣?

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Video Transcript

The components of the vector 𝑢 are negative two, one as the terminal point of the vector is negative two units right (two units left) of the initial point and one unit up from the initial point. What are the components of the vector 𝑣?

A vector on a coordinate grid describes how to move from one point to another. The starting point of the vector is called the initial point. And the endpoint is called the terminal point. The arrow on the vector’s length indicates the direction of the vector. We’re travelling from the initial point to the terminal point.

Vectors in two dimensions have two components. The first describes the horizontal movement to get from the initial point to the terminal point. And the second describes the vertical movement. We define the positive direction to be to the right for horizontal movements and up for vertical movements.

We’ve been given the components of the vector 𝑢 then negative two, one. And the explanation is that the terminal point of this vector is negative two units right and one unit up from the initial point. We can see this on the diagram. Notice that negative two units right means the same as two units left.

Now, let’s have a look to the vector 𝑣. Its initial point is here. And the horizontal and vertical movements have been marked in for us using orange dotted lines. We can see that, to get from the initial point to the terminal point, the horizontal movement is to the left, which means it’s going to be a negative number. Let’s count how many units we need to move, one, two, three, four, five. So we need to move five units to the left, which we can write as negative five units to the right. So the first component of our vector 𝑣 is negative five.

Now, let’s look at the vertical component. And we can see that we have to move up to get from our initial point to the terminal point. So we’re going to have a positive value. Let’s count how many squares we need to move, one, two, three. If we need to move three units up, then the vertical component of our vector 𝑣 will be three, positive three, because up is the positive direction.

We found then that the components of the vector 𝑣 are negative five, three.

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